Russell on Denoting?

This appeared on Daily Nous on 13 September 2022, just a couple days after the expression “the present King of England” changed(?) its meaning(?).

The “To φ or not to φ” comics feature on The Daily Nous is done by Tanya Kostocha ,  Assistant Professor of philosophy at Ashoka University. Russell’s theory of descriptions is long gone, but is still studied for the sake of understanding the variety of refutations and reformulations that succeeded it. Oh, and also for its well-remembered example, “The present King of France is bald” (uttered in 1905, when there was no current King of France).


  1. This seems to be a version of the Sorites paradox, which is about vague terms like heaps and baldness … if one grain of wheat isn’t a heap, and one more doesn’t make a heap, and one more doesn’t and so on grain by grain until you find 1,000,000 grains aren’t a heap.

    Similar with hairs on the head – is Charles bald or not? Where does baldness (or heapage) start? He has a lot of hair around his head but not much on top. In laypersons’ terms he is bald. The fact he is in fact now the present King of England is a bonus.

    A different formulation of the paradox replaces the set of conditional premises with a universal generalization and proceeds by mathematical induction. Let ‘n’ be a variable ranging over the natural numbers and let ‘∀n(…n…)’ assert that every number n satisfies the condition …n…. Further, let us represent the claim ‘For any n, if αn is Φ then αn+1 is Φ’ as ‘∀n(Φαn→Φαn+1)

    Mathematical Induction Sorites



    For example, since a man with 1 hair on his head is bald, and since, for any number of hairs n, if a man with n hairs is bald then so is a man with n+1 hairs, every number n is such that a man with n hairs on his head is bald.

  2. Powers, that would be Bertrand Russell, big name British philosopher of the Twentieth Century. He was important both in academic formal philosophy, and in what is now called public philosophy (opposition to the UK acquiring nuclear weapons, condemning the American war in Vietnam).

    This cartoon is playing off an article where he asked if the statement “The present King of France is bald” would be counted True or False or Neither at a time when there was no King of France.

  3. I have a few hairs growing on top. They began receding 60 years ago. But I’ve never admitted to myself that I’m bald. I’m balding.

  4. Thanks, @Dana K and @narmitaj!

    I thought this was clever because of the timing. Just like “God save the Queen, oh sorry I mean King” and all the Silk baristas who were Q.C. becoming K.C. …. A couple weeks ago the sentence “The present King of England is bald” was problematic in much the way Russell’s version with “France” was. Then because of history marching on, and the promotion of a new King, that same sentence [though a different “utterance” since the time of utterance is different] becomes suddenly non-problematic and simply an empirical question.

    But not without philosophical interest — as Narmitaj reminds us, settling what appears to be simply an empirical question may surprise us with problems. Some coming from Philosophy of Science — what are facts, what are measurements, when do we trust them — plus really old yet unsolved issues, such as a susceptibility to attack by a Heap argument (Sorites)!

  5. While a person who is born, raised, and living in England may refer to Charles (III) as that person’s king, there exists no living person whose title is “King of England” as that position has not existed for hundreds of years. In 1905, there was no living person to refer to as the “King of France” in any truly monarch-ical way (but possibly as respect and deep declaration of loyalty).

    The humor of this “comic” is simply the referral to the mentioned France quote and how the parts of the 2 statements differ in their true/false logic. After that, any discussion is intended to be philosophical; It is a delightful call to the members for discussion.

    @Lark, that was a real day-brightener for me!

    (My apologies for not getting up to find the dictionary today, I am often astonished anew to find there’s no spell-check active in the “Add a Comment” box” even though I’m in Chrome.)

  6. @Deety: no, no, no . . . he and I were both already lost as to the meaning of the comic right when the name ‘Russell’ came up – neither of us realizing it was Bertrand Russell (although come to think on it, Russell Brand might fit your latter category, but that’s getting political, so never mind).

  7. But Kilby, this is only secondarily about the baldness paradox. It’s primarily about the “failed referent” issue — and amusing because that issue “magically” evaporated, for this one particular instance of the very famous example.

    BTW, for the curious, Russell’s answer was that “The present King of France is bald” is False. (Rather than True or No Truth Value.) This came from his analysis of the “definite description” as embodying an existence claim. This was stated very formally — these guys were basically inventing Symbolic Logic — but the informal analysis was that the sentence must be taken as “There is currently an entity which is a King of France, and there is no more than one such, and that [existent and unique] entity is bald.” Because the first clause in that conjunction is False, so is the conjunction. The “[existent and unique]” is an analysis of “the”.

    This was unpalatable to many, including many who were basically his followers, and led to a rich literature. See ]

  8. @Kevin A, thanks for the correction on “King of England”. I was rather surprised to see what’s his name — okay, it’s William — referred to as Prince of Wales, already, as though it is automatic. I recall from The Crown that Charles at the time had a big investiture ceremony, and had to study his Welsh very hard to get thru it.

  9. Oops, I completely missed narmitaj’s comment and I had looked back several times (without refreshing) to see if anyone had touched on philosophy. Could his post have been delayed? (because I’m worried that I might need a trip to the doctor’s : – ))

    And even though I CTRL-X’d and refreshed the comments before posting, Mitch4 got in before me. (thank goodness because it alerted me to the narmitaj post.)

  10. Kevin (and anyone else wondering), yes the comment from Narmitaj was — for reasons unknown — held in Pending, but then released when noticed. As you probably know, WP keeps the original timestamp of such a comment, and then uses that in ordering them on a page.

    But don’t worry, there was nothing about your comment that looks out of place for not mentioning narmitaj’s.

    I forget what Ctrl-X does?

    BTW when you said “After that, any discussion is intended to be philosophical; It is a delightful call to the members for discussion.” it left me unsure if you meant members here or at Daily Nous — the latter since it is a news outlet for academic philosophy and their “membership” moreso than ours may be predisposed to enjoy philosophical chatter.

  11. @MITCH4 (AND ANYONE ELSE WHO MIGHT BE INTERESTED): Line of succession now that Queen E II has died:


  12. @ Mitch – Okay, in addition to to baldness paradox, I find the reference issue tiresome, especially after wasting an hour yesterday on a moronic function in Excel that instead of the simple “zero or one” answer(s) that I needed, produced “one” or a “Value Out of Range” error, forcing me to add superfluous layers of testing. In that vein, the pointer to “Current King of [location]” should have generated “[Nobody]“, so that the “IsBald{}” function would generate a “Missing Argument” error.

    P.S. I suggested the modified “Trecherous Image” caption not just because I think it would have made for a funnier joke, but also because of the “#NotMyKing” protests that have been cropping up in Great Britain, and which have been quashed with an inordinate amount of force and a complete disregard toward the right to free expression.

    P.P.S. Ctrl+X is “cut to clipboard“, the simplest method to pocket an already typed comment while performing a refresh to verify whether an intervening comment has rendered one’s own text obsolete.

  13. Kilby say: In that vein, the pointer to “Current King of [location]” should have generated “[Nobody]“, so that the “IsBald{}” function would generate a “Missing Argument” error.

    Yes, that is one approach! From section 5.1 of that Stanford Encyclopedia article,

    Strawson (1950) objected that Russell’s theory is simply incorrect about the truth conditions of sentences like ‘The present king of France is bald’. According to Russell’s analysis, this sentence is false (since it contains an existence claim to the effect that there is a present king of France), but according to Strawson, this does not conform to our intuitions about the truth of an utterance of that sentence. In Strawson’s view, an utterance of the sentence in a world where there is no present king of France is neither true nor false; perhaps the sentence has a truth value gap, or perhaps it fails to express a determinate proposition (Strawson vacillated on this), but either way it does not appear to be false.

    [He also originated the phrasing of the problem as “failure of a presupposition” which basically led to the entire field of Pragmatics in Linguistics/Philosophy.]

    But read on within that Section 5.1 for later dissents, and these examples:

    Consider the following minimal pairs, where the examples marked with ‘#’ indicate ambivalence about assigning a truth value, and ‘F’ indicates it is more plausible to assign a value of falsehood.

    (34) a. # The present king of France is bald.
    b. F The present king of France is a bald Nazi.
    (35) a. # The present king of France is sitting in a chair.
    b. F The present king of France is sitting that chair.
    (36) a. # The present king of France read Anna Karenina.
    b. F The present king of France wrote Anna Karenina.
    (37) a. # The present king of France heard about Goldbach’s conjecture.
    b. F The present king of France proved Goldbach’s conjecture.

  14. Ctrl-X is the ‘cut’ of ‘cut-and-paste’ (in Windows, command-x on an Apple Mac). In an editable document window, it copies the selected text and deletes it from the document. I could use Ctrl-C (copy) to preserve my un-posted comment before refreshing the comments; I just like to know for sure that no glitch will cause it to posted.

    I mostly meant the Daily Nous group and/or the Worlds loosely-knit philosophy fans , but I actually forgot that I knew (momentarily) the name of that site; perhaps because all my mind had wanted to do was try to think of more possibilities of what “nous” could be. (I had 7 years of French, so I feel bias may be blocking me.) (I believe the only ones are ‘news’ and ‘us’.)

  15. And I never was sure how to pronounce it!

    Here’s Merriam-Webster:

    ˈnüs also ˈnau̇s : MIND, REASON: such as
    an intelligent purposive principle of the world
    the divine reason regarded in Neoplatonism as the first emanation of God
    ˈnau̇s chiefly British : COMMON SENSE, ALERTNESS

  16. I don’t know why I find this concept of succession so fascinating: Is it ’cause I read so many British crime books and watch so many BritComs and BritCrime series? Is it ’cause the entire concept of a monarchy – no matter what country (and I come from a country where there is a very wealthy monarchy) is so anachronistic and ludicrous? Or because so much of world history derives from so-called legitimate and illegitimate monarchies and wars between them (and their interconnectedness with wars fought over/between religions)? All I know is that I am more interested in all this monarchical nonsense than is a friend of mine who was born in Scotland and has lived in England for much of her life ‘-)

    I’m interested also in this #NotMyKing news . . . will have to look that up. Probably taken from the #NotMyPresident meme we’ve used in the US.

  17. That last bit reminds me of the research that found that people will be more sure of the truth of a proposition when you add information, when statistically, adding another condition should make it less likely to be true. I can’t remember it exactly, but something like, “Bob is an accountant” people go, meh, maybe, but “Bob is an accountant who likes to collect buttons” and people go, yeah, that’s probably true, even though statistaclly you’d have to multiply the likelihood of either being true, making both being true more unlikely than either in isolation.

    Anyone have links to what I’m talking about?

  18. @Andréa – The reason I thought that there was no spell-check was my typing two words I was sure couldn’t be spelled the way I spelled them. The only one I remember is “debuted”. I also tried to misspell a word or two and they weren’t getting highlighted. (I could have misspelled them into words I didn’t know existed.)

    I am glad that I had enough conviction to at least finish typing “debuted” before it became so unfamiliar- feeling that I decided to change the sentence.

  19. Whoops, fast moving discussion! I meant, the last bit of Mitch’s quoting section 5.1 of that Stanford Encyclopedia article…

  20. @larK, I think it’s in the same ballpark as that “added info” study, but only example 34 is really close to that — the others are more like, we have additional basis for knowing this claim must be false regardless of the success or failure of the referring-expression. Thus in 35 (which I think has a typo, missing “in” in b.) we can see that nobody at all is sitting in that chair; in 36 we know who wrote Anna Karenina and he wasn’t in any way the King of France; in 37 we know that nobody has proven Goldbach’s conjecture.

  21. @Mitch4 – re: ‘nous’ WOW!.. I took a look in the Oxford Dictionary of English. I can’t find any memory in my mind about knowing that word. I feel there must have been several movies (British) where I missed it. (hard to ignore it in a book; perhaps easy to forget)

  22. @Mitch4: Charles was named as Prince of Wales in 1958, but they didn’t hold his investiture until 1969. The investiture ceremony is what he studied Welsh for.

    While William has been named Prince of Wales already, it is not known if he will have an investiture, nor when that might take place if it does.

  23. The discussion reminds me a bit of the Brazilian Olympiad Pinocchio’s Hats logic puzzle.

    Assume that both of the following sentences are true:

    • Pinocchio always lies;
    • Pinocchio says, “All my hats are green.”

    We can conclude from these two sentences that:

    (A) Pinocchio has at least one hat.
    (B) Pinocchio has only one green hat.
    (C) Pinocchio has no hats.
    (D) Pinocchio has at least one green hat.
    (E) Pinocchio has no green hats.

    In case it’s not clear, only one answer is correct.

  24. LOTS of money the British taxpayers are putting out – funeral, coronation, investiture. Does the family itself pay for ANY of this??

  25. Yes, they do; the British tax payer pays close to nothing to the royal family (you can argue about extra police support and army and navy support). Remember, the king WAS the State, all money was his; one of the Georges agreed to give most of it to a fund the government could administer so he didn’t have to bother with paying for things like parliament, but that money remains the royals money. They get paid out of that fund (about 25% currently, but that is rather higher than normal because they are renovating some castles). The monarch also has two duchies they still own outright, that provide about a £20 million a year income — the queen chose to voluntarily pay taxes on this, so that’s a boon to the tax payer. (Remember, the king is the State, so when you pay taxes to the state, you were paying them to the king — so who does a king pay taxes to? Himself?)

  26. Brian, I saw “hat” and thought it was going to be one of that family of puzzles involving a group of people and rules about who can see the others’ hats, and deducing one’s own hat color based on “A didn’t announce a solution but he would have if he was seeing two white hats so he must be seeing …” .

    But this is a good riddle of a different flavor. The nature of hats is not a factor as with the indirect-reasoning puzzles.

    I briefly seized on (C) but of course that is wrong, and based on a too-quick misreading. The right answer has to be (A). This turns on the modern principle of a universal being satisfied vacuously, but still counted true. At one time an All implied an Exists, but not any more.

    BTW have you at some point been a fan of Raymond Smullyan?

  27. In one of Martin Gardner’s books, ther was a comment that most “lying” puzzles and paradoxes depend on an unrealistic (far too logical) understanding of the art of lying. If “all my hats are green” is to be interpreted as a “logical” falsehood, then it could mean “not all of my hats are green” or it could mean “all of my hats are not green” (hence the determination that “a” is correct: he has at least one hat, since the latter interpretation does not rule out “all=one”). However, if the lie is an “artful” or “denotational” lie, then the statement might mean “all of my ‘hats’ are white wigs“, in which case both C and E would be correct. The condition “only one answer is correct” could then be interpreted as ruling out the art of lying.

  28. Vaguely relevant: the queue to see the Queen lying in state has reached capacity (about five miles long) and so the ability to join the end of it has been paused for six hours.

    The question is (without researching it): before people are again allowed to join the end of the queue, what are they doing now? Are they milling about randomly at a distance, constantly being shooed away by officers if they get too close to the end? Or are they actually queueing unofficially to join the end of the queue – in which case, aren’t they still actually queueing to see the Queen in some way?

  29. Similarly, if the true statement is “all of my hats were green yesterday”, then they still might be green today, or not, in which case we can’t conclude any of A-E at all and any of them might or might not be true, depending on whether Pinocchio was visited by a hat thief overnight, and if so whether he’s a victim or a fence.

    A more subtle variation on that is if what he means is that the hats are green now but subject to change, since that is not the expected interpretation of “are” or the one we’re using in points A-E, and in that case all of A-E would be false.

    Somewhat relatedly, also, deducing anything at all from false statements tends to rely on the excluded middle, and that’s not necessarily sustainable here. In particular, whether a hat is green or not is probably vulnerable to a sorites argument. Similarly, if Pinocchio is out to make trouble, he might have acquired a green hat using a Turing-complete contract such that “mine” is undecidable.

    (In the traditional interpretation, btw, we know that the hat he must have isn’t green, because the negation of “for all h in hats, Mine(h) implies Green(h)” is “exists h in hats, Mine(h) and not Green(h)”. Or “for all h, Hat(h) implies Mine(h) implies Green(h) ” becomes “exists h, Hat(h) and Mine(h) and not Green(h)”.)

  30. Related to the green hats is the “confirming instance” paradox. If you claim “all hats are green” and go out to observe hats, every green hat you find strengthens your position … until you find a hat that is not green, of course.

    Now if you set out to prove “All crows are black,” you can go out counting crows. But at the end of the day, you might have seen only a few crows. Instead, consider that the statement “All crows are black” is equivalent to “all things that are not black are not crows.” Now you just look around: there’s a green hat! That’s not black, and it’s not a crow. You can find thousands of confirming instances just looking around your room.

  31. Mark in Boston, I heard about that under the name Hempel’s Paradox or the Ravens Paradox. I don’t remember learning any good answers to it!

    Though it’s not really a logical paradox. It’s more a trick-argument, for the position “verification as increasing probability is hoooey”.

    Or else “you can’t just go around swapping in contrapositives”

  32. The book by Martin Gardner† that I mentioned in commenting about the “art of lying” also contains an excellent description of Hempel’s paradox. MiB’s description is similar enough that I wonder whether he has read the same book.

    @ Mitch – According to Gardner, there are several possible reasons that the logical reversal seems to defy “common sense”. For instance, MiB’s “green hat” example is a potential confirmation for “all crows are black“, but it also could be used to confirm “all crows are white“. In addition, serious problems arise when the set of objects is effectively unlimited (both “crows” and “objects in the room” would be exceedingly difficult to enumerate). Gardner provides a counter example with a more limited set, namely “all the typists employed by a (large) company”, and examines the intersection of the subsets “redhead” and “married” (it’s an old book, the implicit sexism was “normal” back when it was written).

    P.S. † “The Scientific American Book of Mathematical Puzzles and Diversions” (1959) — I received my copy (of the 3rd printing) as a gift in 1985. The discussion of Hempel’s paradox starts on page 52, the “lying” puzzle is on page 25, but the best part are the solutions (and a hilarious letter from a reader) on pages 28-32.

  33. Ah yes, I followed his column in the magazine for a long time, and also enjoyed many of the collections in book form.

    I see that there is now “Martin Gardner’s Mathematical Games: The Entire Collection of his Scientific American Columns”. This link is to an e-book listing: . This doesn’t seem to be a traditional printed book. Amazon is featuring a CDR, of all things!

    But I see they have reissued some of the collections as books. “Hexaflexagons and Other Mathematical Diversions: The First ‘Scientific American’ Book of Puzzles and Games Paperback – September 15, 1988” That date is certainly not the original publication — I made some hexaflexagons that my family took delight in playing with while sitting in the Florida room watching tv, in a house that we lived in in the first half of the sixties. And I made a Science Fair poster project on “Topology” based on a couple chapters (or not-yet-collected magazine columns by Gardner.

    I was reminded that Gardner spent some time at the University of Chicago, whose neighborhood I live in. I don’t remember the name of a character he assumed for writing a mx of genuine number theory and amusing crank numerology. But this Dr. oh oh oh it was Dr Matrix! This Dr. Matrix once remarked on what look like unlikely-by-chance or supernatural-influenced street addresses; was Bantam Books at 666 Fifth Avenue a satanic enterprise, for instance.? Then Gardner in propria persona mentions 1234 East 56th Street in Chicago and some of the notable occupants of that building. (Such as the Institute of General Semantics, which still exists but not at that address.) Then I realized I walked by that spot very regularly, and I checked that the street address remained. Though it had been converted from offices to a Co-op or Condo building, with large apartments very suitable for renting to a group of like a half-dozen students.

  34. At some point I discovered that I had started reading Martin Gardner shortly after I learned to read. This was in the 1950’s, and my parents bought me a subscription to Humpty Dumpty’s Magazine for Children. “Humpty Dumpty sat on a wall. Humpty Dumpty had a great fall. All the King’s horses and all the King’s men Couldn’t put Humpty together again. But an American doctor with scissors and glue Put Humpty together as good as new. And now he is back on the scene Busily editing this magazine.”

    Supposedly Humpty Dumpty was the editor, but Gardner was one of the writers and editors. He wrote The Adventures of Humpty Dumpty Junior, a little egg, Humpty Senior’s son. The magazine was full of little puzzles and tricks and clever cut-outs.

    I read everything by Martin Gardner that I could get my hands on. And made lots of hexaflexagons.

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